I'll apologize in advance if this is not an appropriate place for my question. My background is not in physics, and my understanding of quantum mechanics is extremely rudimentary at best, so I hope you'll be forgiving of my newbish question.

Given a system of entangled particles (eg, 2 or more electrons), possibly in a superposition state: if the particles interact with each-other, what effect does this have on their quantum state? Is their state now determined (but perhaps unknown until observed)?

Interactions within the system could lead to entangled or disentangled states, as @Lagerbaer said in his answer. However, this has nothing to do with decoherence (mentioned in the title), which can only be caused by interaction with another (external) system.
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JoeJan 14 '12 at 11:16

What if we had ferromagnetic coupling? Then the ground-state is degenerate. It could be the state above but with a + sign in the superposition, which would again be an entangled state, or it could be either $|\uparrow \uparrow\rangle$ or $|\downarrow \downarrow\rangle$ and these two states are not entangled.

So interactions "with each other" can either lead to an entangled or disentangled state.

It's the other way around: interactions within the system will generally result in an entangled state, while interactions with the environment are associated with measurement and decoherence/collapse.

Although it's possible in principle for an interaction within the system to disentangle it, it won't generally happen because unentangled/separable states are very special (they are a subspace of measure 0 of the space of multiparticle states). On the other hand, independently measuring parts of the system always breaks the entanglement and leave you with a separable state (a partially separable state if it's a partial measurement).